Revista Matemática Complutense

, Volume 28, Issue 3, pp 549–597 | Cite as

The convenient setting for Denjoy–Carleman differentiable mappings of Beurling and Roumieu type

  • Andreas Kriegl
  • Peter W. Michor
  • Armin Rainer


We prove in a uniform way that all Denjoy–Carleman differentiable function classes of Beurling type \(C^{(M)}\) and of Roumieu type \(C^{\{M\}}\), admit a convenient setting if the weight sequence \(M=(M_k)\) is log-convex and of moderate growth: For \(\mathcal C\) denoting either \(C^{(M)}\) or \(C^{\{M\}}\), the category of \(\mathcal C\)-mappings is cartesian closed in the sense that \(\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)\) for convenient vector spaces. Applications to manifolds of mappings are given: The group of \(\mathcal C\)-diffeomorphisms is a regular \(\mathcal C\)-Lie group if \(\mathcal C \supseteq C^\omega \), but not better.


Convenient setting Denjoy–Carleman classes of Roumieu and Beurling type Quasianalytic and non-quasianalytic mappings of moderate growth Whitney jets on Banach spaces 

Mathematics Subject Classification

26E10 46A17 46E50 58B10 58B25 58C25 58D05 58D15 


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Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  • Andreas Kriegl
    • 1
  • Peter W. Michor
    • 1
  • Armin Rainer
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria

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